Density theorems for Riemann's auxiliary function
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917700545019904 |
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| author | de Reyna, Juan Arias |
| author_facet | de Reyna, Juan Arias |
| contents | We prove a density theorem for the auxiliar function $\mathop{\mathcal R}(s)$ found by Siegel in Riemann papers. Let $α$ be a real number with $\frac12< α\le 1$, and let $N(α,T)$ be the number of zeros $ρ=β+iγ$ of $\mathop{\mathcal R}(s)$ with $1\ge β\geα$ and $0<γ\le T$. Then we prove \[N(α,T)\ll T^{\frac32-α}(\log T)^3.\]
Therefore, most of the zeros of $\mathop{\mathcal R}(s)$ are near the critical line or to the left of that line. The imaginary line for $π^{-s/2}Γ(s/2)\mathop{\mathcal R}(s)$ passing through a zero of $\mathop{\mathcal R}(s)$ near the critical line frequently will cut the critical line, producing two zeros of $ζ(s)$ in the critical line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14987 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Density theorems for Riemann's auxiliary function de Reyna, Juan Arias Number Theory Primary 11M06, Secondary 30D99 We prove a density theorem for the auxiliar function $\mathop{\mathcal R}(s)$ found by Siegel in Riemann papers. Let $α$ be a real number with $\frac12< α\le 1$, and let $N(α,T)$ be the number of zeros $ρ=β+iγ$ of $\mathop{\mathcal R}(s)$ with $1\ge β\geα$ and $0<γ\le T$. Then we prove \[N(α,T)\ll T^{\frac32-α}(\log T)^3.\] Therefore, most of the zeros of $\mathop{\mathcal R}(s)$ are near the critical line or to the left of that line. The imaginary line for $π^{-s/2}Γ(s/2)\mathop{\mathcal R}(s)$ passing through a zero of $\mathop{\mathcal R}(s)$ near the critical line frequently will cut the critical line, producing two zeros of $ζ(s)$ in the critical line. |
| title | Density theorems for Riemann's auxiliary function |
| topic | Number Theory Primary 11M06, Secondary 30D99 |
| url | https://arxiv.org/abs/2406.14987 |